- Distance Between Two Skew Lines Figure 2 73 Industrial Pipe Installations Often Feature Pipes Running In Different Direc 1 (158.59 KiB) Viewed 74 times
They need to be comlpleatly ansered.
Distance between Two Skew Lines Figure 2.73 Industrial pipe installations often feature pipes running in different directions. How can we find the distance between two skew pipes? Finding the distance from a point to a line or from a line to a plane seems like a pretty abstract procedure. But, if the lines represent pipes in a chemical plant or tubes in an oil refinery or roads at an intersection of highways, confirming that the distance between them meets specifications can be both important and awkward to measure. One way is to model the two pipes as lines, using the techniques in this chapter, and then calculate the distance between them. The calculation involves forming vectors along the directions of the lines and using both the cross product and the dot product. The symmetric forms of two lines, L1 and L2, are L, : x - 110 1 XX1 y-yi-2-21 L2: ***2 – Y72=2622 b2 C2 You are to develop a formula for the distance d between these two lines, in terms of the values a, b, C1; a2, b2, C2; x1, yj, 21; and x2, Y2, 22. The distance between two lines is usually taken to mean the minimum distance, so this is the length of a line segment or the length of a vector that is perpendicular to both lines and intersects both lines. 1. First, write down two vectors, V, and V2, that lie along L1 and L2, respectively. 2. Find the cross product of these two vectors and call it N. This vector is perpendicular to V, and v2, and
Chapter 2 | Vectors in Space 205 hence is perpendicular to both lines. 3. From vector N, form a unit vector n in the same direction. 4. Use symmetric equations to find a convenient vector V12 that lies between any two points, one on each line. Again, this can be done directly from the symmetric equations.